Optimal. Leaf size=80 \[ \frac {16 a^2 \left (a x^2+b x^3\right )^{5/2}}{315 b^3 x^5}-\frac {8 a \left (a x^2+b x^3\right )^{5/2}}{63 b^2 x^4}+\frac {2 \left (a x^2+b x^3\right )^{5/2}}{9 b x^3} \]
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Rubi [A] time = 0.13, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2016, 2014} \begin {gather*} \frac {16 a^2 \left (a x^2+b x^3\right )^{5/2}}{315 b^3 x^5}-\frac {8 a \left (a x^2+b x^3\right )^{5/2}}{63 b^2 x^4}+\frac {2 \left (a x^2+b x^3\right )^{5/2}}{9 b x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 2014
Rule 2016
Rubi steps
\begin {align*} \int \frac {\left (a x^2+b x^3\right )^{3/2}}{x} \, dx &=\frac {2 \left (a x^2+b x^3\right )^{5/2}}{9 b x^3}-\frac {(4 a) \int \frac {\left (a x^2+b x^3\right )^{3/2}}{x^2} \, dx}{9 b}\\ &=-\frac {8 a \left (a x^2+b x^3\right )^{5/2}}{63 b^2 x^4}+\frac {2 \left (a x^2+b x^3\right )^{5/2}}{9 b x^3}+\frac {\left (8 a^2\right ) \int \frac {\left (a x^2+b x^3\right )^{3/2}}{x^3} \, dx}{63 b^2}\\ &=\frac {16 a^2 \left (a x^2+b x^3\right )^{5/2}}{315 b^3 x^5}-\frac {8 a \left (a x^2+b x^3\right )^{5/2}}{63 b^2 x^4}+\frac {2 \left (a x^2+b x^3\right )^{5/2}}{9 b x^3}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 47, normalized size = 0.59 \begin {gather*} \frac {2 x (a+b x)^3 \left (8 a^2-20 a b x+35 b^2 x^2\right )}{315 b^3 \sqrt {x^2 (a+b x)}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 4.66, size = 70, normalized size = 0.88 \begin {gather*} \frac {2 \left (x^2 (a+b x)\right )^{3/2} \left (63 a^2 (a+b x)^{5/2}+35 (a+b x)^{9/2}-90 a (a+b x)^{7/2}\right )}{315 b^3 x^3 (a+b x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 62, normalized size = 0.78 \begin {gather*} \frac {2 \, {\left (35 \, b^{4} x^{4} + 50 \, a b^{3} x^{3} + 3 \, a^{2} b^{2} x^{2} - 4 \, a^{3} b x + 8 \, a^{4}\right )} \sqrt {b x^{3} + a x^{2}}}{315 \, b^{3} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.24, size = 173, normalized size = 2.16 \begin {gather*} -\frac {16 \, a^{\frac {9}{2}} \mathrm {sgn}\relax (x)}{315 \, b^{3}} + \frac {2 \, {\left (\frac {21 \, {\left (3 \, {\left (b x + a\right )}^{\frac {5}{2}} - 10 \, {\left (b x + a\right )}^{\frac {3}{2}} a + 15 \, \sqrt {b x + a} a^{2}\right )} a^{2} \mathrm {sgn}\relax (x)}{b^{2}} + \frac {18 \, {\left (5 \, {\left (b x + a\right )}^{\frac {7}{2}} - 21 \, {\left (b x + a\right )}^{\frac {5}{2}} a + 35 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{2} - 35 \, \sqrt {b x + a} a^{3}\right )} a \mathrm {sgn}\relax (x)}{b^{2}} + \frac {{\left (35 \, {\left (b x + a\right )}^{\frac {9}{2}} - 180 \, {\left (b x + a\right )}^{\frac {7}{2}} a + 378 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{2} - 420 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{3} + 315 \, \sqrt {b x + a} a^{4}\right )} \mathrm {sgn}\relax (x)}{b^{2}}\right )}}{315 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 46, normalized size = 0.58 \begin {gather*} \frac {2 \left (b x +a \right ) \left (35 b^{2} x^{2}-20 a b x +8 a^{2}\right ) \left (b \,x^{3}+a \,x^{2}\right )^{\frac {3}{2}}}{315 b^{3} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.55, size = 53, normalized size = 0.66 \begin {gather*} \frac {2 \, {\left (35 \, b^{4} x^{4} + 50 \, a b^{3} x^{3} + 3 \, a^{2} b^{2} x^{2} - 4 \, a^{3} b x + 8 \, a^{4}\right )} \sqrt {b x + a}}{315 \, b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.18, size = 47, normalized size = 0.59 \begin {gather*} \frac {2\,\sqrt {b\,x^3+a\,x^2}\,{\left (a+b\,x\right )}^2\,\left (8\,a^2-20\,a\,b\,x+35\,b^2\,x^2\right )}{315\,b^3\,x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (x^{2} \left (a + b x\right )\right )^{\frac {3}{2}}}{x}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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